Extraction of fetal cardiac signals

ABSTRACT

A method for processing cardiac signals includes accepting, from a sensor system, a set of one or more signals, the signals including components of a desired cardiac signal and components of a substantially periodic interfering signal. The method is applicable for extraction of desired fetal cardiac signals from signals with interference from the maternal cardiac signal. A periodicity of the interfering signal is determined, and one or more iterations of mitigating an effect of a component of the signals that exhibit periodicity at the determined periodicity of the interfering signal are performed. In some examples, the method further includes determining a periodicity of the desired cardiac signal, and performing one or more iterations of enhancing an effect of a component of the signals that exhibit periodicity at the determined periodicity of the desired cardiac signal.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application claims priority to U.S. Provisional Application Ser.No. 61/116,870, filed Nov. 21, 2008, the contents of which areincorporated herein by reference.

BACKGROUND

This invention relates to extraction of fetal cardiac signals fromsignals acquired from a subject.

Monitoring of fetal cardiac signals can be difficult due to the presenceof both maternal and fetal cardiac signals in raw signals acquired froma subject, as well as a relatively low fetal signal level relative tothe maternal signal and other noise sources. One approach to acquiringfetal cardiac signals involves the use of electrodes, such as surfaceelectrodes positioned on the maternal abdomen, which may provide lowersignal to noise levels as compared to invasive approaches to acquiringfetal signals (such as using fetal scalp electrodes).

SUMMARY

In a general aspect, a method for processing signals includes accepting,from a sensor system, a set of one or more signals. The signals includecomponents of a desired signal and components of a interfering signalhaving a periodicity. The method further includes determining, by aprocessing engine, the periodicity of the interfering signal; andperforming, by the processing engine, one or more iterations ofmitigating an effect of a component of the signals that exhibitsperiodicity at the determined periodicity of the interfering signal toform a processed signal.

Embodiments may include one or more of the following. The methodincludes providing the processed signal to a display system.

The interfering signal includes at least one of a substantially periodicand a pseudo periodic signal. Determining the periodicity of theinterfering signal includes determining a time varying period of saidsignal. The method includes determining a periodicity of the desiredsignal and performing one or more iterations of enhancing an effect of acomponent of the signals that exhibits periodicity at the determinedperiodicity of the desired signal.

The desired signal is a cardiac signal. The interfering signal includesa second cardiac signal. The periodicity of the interfering signalcomprises a quantity characterizing a heartbeat period of the secondcardiac signal. The quantity characterizing the heartbeat period of thesecond cardiac signal comprises an estimated R-R interval. The desiredcardiac signal comprises a fetal cardiac signal and the interferingsignal comprises a maternal cardiac signal. The desired cardiac signalcomprises a cardiac signal of a first fetus and the interfering signalcomprises a cardiac signal of a second fetus. The interfering signalfurther includes a maternal cardiac signal.

Mitigating the effect of a component of the signals that exhibitperiodicity at the determined periodicity of the interfering signalincludes identifying a relative contribution of the component of thesignals that exhibit periodicity at the determined periodicity; andprocessing the signals according to the identified relativecontribution. Identifying the relative contribution of the component ofthe signals that exhibit periodicity at the determined periodicityincludes computing a generalized eigenvalue based on correlation of thesignals at the determined periodicity of the interfering signal.Processing the signals according to the identified relative contributionincludes applying a Bayesian filtering approach.

In another general aspect, a cardiac monitoring system includes an inputsubsystem for accepting signals. The signals include components of adesired signal and components of an interfering signal having aperiodicity. The cardiac monitoring system further includes a dataprocessing system for processing the accepted signals. The dataprocessing system is configured to perform the steps of determining theperiodicity of the interfering signal, and performing one or moreiterations of mitigating an effect of a component of the signals thatexhibits periodicity at the determined periodicity of the interferingsignal. The cardiac monitoring system also includes a data outputsubsystem for providing information determined from the processedsignals.

Embodiments may include one or more of the following. The desired signalis a cardiac signal.

In a further general aspect, software embodied on a computer-readablemedium includes instructions for causing a data processing system toaccept a set of one or more signals, the signals including components ofa desired signal and components of a interfering signal having aperiodicity; determine the periodicity of the interfering signal; andperform one or more iterations of mitigating an effect of a component ofthe signals that exhibits periodicity at the determined periodicity ofthe interfering signal.

In another general aspect, a method for processing a input signal thatincludes a pseudo-periodic signal present with another signal includesaccepting the input signal. The input signal is representable as avector signal in a multidimensional vector space. The method furtherincludes forming a phase signal representing a time-varying periodicityof the pseudo-periodic signal; identifying a subspace of the vectorspace that exhibits correlation at the time-varying periodicity; forminga component of the input signal according to the identified subspace;and processing the input signal to either enhance or to mitigate andeffect of the formed component to produce a processed signal.

Embodiments may include one or more of the following. Accepting theinput signal includes accepting a biosignal. The pseudo-periodic signalhas a periodicity related a periodic biological process. The biosignalincludes a signal acquired by sensing a signal from a human body, andthe periodic biological process includes a cardiac process.

The method further includes presenting the processed signal on adisplay. Forming the phase representation includes identifyingrepetitions of a signal characteristic in the input signal. Identifyingthe subspace includes determining a generalized eigenspace from acorrelation of the input signal at a time varying lag determined fromthe phase representation and from a correlation of the input signal atzero lag. Forming the component of the input signal according to theidentified subspace includes projecting the input signal onto theeigenspace. Processing the input signal to either enhance or to mitigatean effect of the formed component to produce a processed signal includesat least one of adding or subtracting the projection of the signal ontothe eigenspace.

The method for extraction of fetal cardiac signals described herein hasa number of advantages. In general, the method enhances a weak fetalsignal that is overshadowed by strong maternal electrocardiogram (ECG)or respiration interference. The retrieved fetal signal preservesclinical information such as signal morphology and themulti-dimensionality of the signal. For instance, the multi-dimensionalnature of the source (e.g., the fetal heart) is preserved, facilitatingdiagnosis along more than one axis and allowing determination of theangle of the fetal cardiac dipole and the presentation or movement ofthe fetus along the fetal body axis. A multi-dimensional dipolereconstruction also allows a Dower-like transform to be performed toreconstruct the surface map of the mother's abdomen and to assessmetrics such as QT dispersion. Referring to FIG. 13, standard clinicalmetrics of an ECG 350, such as QT interval 352 and ST elevation 354, canalso be evaluated using the extracted fetal signals. Furthermore,localization of the R-peak 356 of a fetal ECG allows for accurateevaluation of heart rate variability (HRV) metrics.

Cardiac signals can be analyzed online in real time or offline after ameasurement. Real-time fetal ECG analysis provides the opportunity toobserve the fetal response to inter-uterine contractions and to identifyabnormalities in the fetal ECG (e.g., arrhythmias and QT lengthening)associated with such stress.

The measurement and analysis can be used at any time during pregnancy,and does not depend on fetal presentation or gestational age. The methodcan be used prior to or during labor. Multiple fetuses can be monitoredsimultaneously and separated for individual diagnosis. The analysis isrobust against movement of the fetus or the electrodes. The measurementsare noninvasive, thus posing little or no risk to the mother or thefetus. Simple monitoring equipment can be used, allowing for inexpensiveand relatively quick measurements.

The analysis is applicable to degenerate mixtures of maternal and fetalcardiac signals and noise without any prior assumptions about thedimensionality of the maternal or fetal cardiac signal subspaces.Mixtures of signals having fewer numbers of channels than expectedsources can be evaluated without losing the important clinical fetalcardiac components during denoising. Maternal interference and othernoise can be eliminated by using even a small number of recordingchannels and without reducing the rank of the observations during thedenoising procedure. At most, a single reference channel (e.g., amaternal reference channel) is used, but the reference channel may berecorded from any arbitrary channel and is not constrained to bemorphologically similar to the contaminating waveform.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is an illustration of the collection of fetal abdominalelectrocardiogram (ECG) from multiple electrodes.

FIG. 2 is a block diagram of a baseline wander removal system.

FIG. 3 is a flow chart of a filtering scheme.

FIG. 4A is a graph of an exemplary cardiac signal and cardiac phasesignal.

FIG. 4B is a polar representation of an exemplary ECG using the ECGphase signal.

FIGS. 5A and 5B are block diagrams of an iterative subspace removalscheme.

FIG. 6 is a flow chart of iterative subspace decomposition.

FIGS. 7A and 7B are flow charts of a scheme for extraction of a fetalcardiac waveform.

FIG. 8A is a graph of an ECG dataset.

FIG. 8B is a graph of maternal ECG components extracted from the datasetof FIG. 8A.

FIG. 8C is a graph of fetal ECG components extracted from the dataset ofFIG. 8A.

FIG. 8D is a graph of maternal and fetal ECG components extracted fromthe dataset of FIG. 8A.

FIG. 9A is a graph of an ECG dataset.

FIGS. 9B-9F are graphs showing the results of 1-5 iterations,respectively, of removal of the maternal subspace from the dataset ofFIG. 9A

FIG. 10 is a graph showing the results of a secondary deflation stepapplied to the results of the last iteration shown in FIG. 9F.

FIG. 11 is a graph comparing an averaged fetal ECG beat from a scalpelectrode and an average beat extracted from an abdominal ECG during thesame time period.

FIG. 12 is a graph showing the distribution of the difference in fetalST deviation between ECG obtained from a fetal scalp monitor and ECGextracted from maternal abdominal signals.

FIG. 13 is a graph of an exemplary ECG.

DETAILED DESCRIPTION

Noninvasive monitoring of fetal cardiac signals is often made difficultby noise and interference from the stronger maternal cardiac signal. Inorder to analyze the fetal cardiac signal, signals recorded from thebody surface of a pregnant woman are processed to obtain a purifiedfetal cardiac signal. Maternal cardiac artifacts, which are the dominantsource of interference in fetal signals, are removed from noisy signalmixtures. After the maternal cardiac interference is removed, dependingon the signal quality and fetal gestational age, the quality of thefetal cardiac signal can be further improved in a post-processing step.Signal quality of the ECG electrode recordings can be assessed bycomparing temporal and spectral coherence between recorded channels.

The extracted fetal signal is multidimensional (i.e., it represents morethan a single quantity, such as heart rate, or a single ECG channel) andcontains diagnostic cardiac information pertaining to the health of thefetus or fetuses carried by the mother. For instance, the presentationof the fetus can be determined from the multidimensional fetal cardiacsignal. Standard clinical metrics related to the morphology of the fetalcardiac signal are also accessible.

The term “cardiac signal” as used herein refers to any waveformcontaining morphological characteristics of a cardiac signal, such as anelectrocardiogram (ECG) or magnetocardiogram (MCG). In some cases, thetechniques described herein are described with reference to the ECG;however, these techniques apply equally to the MCG or any similarmodalities of cardiac monitoring.

1 DATA COLLECTION

From a signal processing perspective, the heart can be considered as adistributed signal source with infinite dimensions. Depending on thedistance between the heart (either the maternal heart or the fetalheart) and electrodes positioned on the body surface, the electrical orelectromagnetic activity of the heart can be approximated as adistributed source with a lower rank subspace. The study of thecomponents of such subspaces provides valuable clinical informationabout the vectorcardiograph (VCG) representation of the cardiacwaveforms and the well-being of the heart. To obtain a VCGrepresentation of cardiac waveforms of an adult, signals are recordedfrom different positions on the body, with each signal containingspecific information about the cardiac activity represented in uniquemorphology (shape) and frequency content. However, since the fetus isnot directly accessible from more than one angle (and even then onlyduring labor), fetal VCG representation is only achievable by signalprocessing techniques.

Referring to FIG. 1, a noninvasive system 100 for fetal cardiac signalextraction makes use of signals representative of an ECG of a fetus 102and an ECG of a pregnant woman 104. The signals are acquired from a setof signal electrodes 106 distributed over the maternal abdomen 108and/or lower back 110 and thus far from the maternal heart. Arepresentative signal 112 acquired by one of the signal electrodes 106shows that the signal is dominated by maternal ECG, although fetal ECGis also present. In some examples, the fetal heartrate may be determinedusing other sensing modes such as ultrasound or the use of a fetal scalpelectrode. However, the full representation of the ECG is moreinformative than these techniques.

In some embodiments, only one maternal abdominal sensor is used.However, for more detailed studies, the use of multiple sensorswell-distributed over the maternal body reduces the risk of degeneratemixtures of signals and improves the quality of the fetal signalextraction. This is because a more accurate representation of thematernal ECG allows it to be removed more effectively from the mixture.For instance, between six and sixteen signal electrodes 106 may be usedto retrieve the fetal waveform morphology, although fewer electrodes canbe used when the fetal ECG is strong. Even more electrodes may be usedfor dynamic reselection of channels containing a stronger fetal signalcontribution, especially during long monitoring sessions when the fetusmay move.

The acquired signals also include one or more reference signals thatprovide information about the maternal heart. These signals may includesignals from a maternal reference channel associated with a set of oneor more maternal reference electrodes 114 placed on the woman's chest116 near the heart from which a reference maternal cardiac signal 118can be determined. Alternatively, maternal cardiac electrodes may beplaced on the navel, hand, or leg of the pregnant woman. In someembodiments, a reference signal is obtained from a signal acquired byone of the signal electrodes 106 and which has a dominant maternalheartbeat. In other examples, other sensing modes are used to determinethe maternal heartrate, for example, using one or more of ultrasound,imaging, bloodpressure sensing. The maternal reference channel providesinformation regarding the maternal heartrate, for example, based onmaternal fiducial point or beat (R-peak) detection. The morphology ofthe maternal reference channel is not constrained to match that of thecontaminating maternal signal recorded by the abdominal signalelectrodes 106.

In general, the data processing methods described below are notsensitive to the sampling rate of the maternal and fetal ECG, providedthe waveforms are not disturbed. However, data sampling at 500 Hz-1000Hz, quantized with 14-16 bits, generally enables a high quality fetalECG signal to be obtained. Higher sampling rates may also be used butrequire a higher processing load.

2 DATA PROCESSING

The acquired signals are treated as a vector, time-sampled, zero-meansignal x(t). That is, if there are N electrodes, the signal isrepresented as an N-dimensional vector. The vector signal is modeled ashaving three components:

x(t)=x _(f)(t)+x _(m)(t)+x _(n)(t),

where x_(f) (t) is a desired N-dimensional component of the signalrepresenting the fetal cardiac signal (without any reduction indimensionality from the originally acquired signals), x_(m)(t) is thematernal interference component, and x_(n)(t) is the noise component.Although represented as N-dimensional signals, some or all of the signalor noise components are not necessarily full rank (i.e., their variationmay be confined to a lower-dimensional subspace).

The maternal and fetal signals, x_(m)(t) and x_(f)(t), respectively, areeach characterized by a pseudo-periodic structure that is repeated inall the channels synchronous with the maternal and fetal heart beats,respectively. The difference in the rates of the maternal and fetalsignals enables a separation of the signals as described further below.

In general, the methods described herein identify and remove anyperiodic structure in the acquired signals that is synchronous with thereference maternal ECG R-peaks. That is, the maternal signal is removedby determining the maternal heartrate and removing components of therecorded signal that exhibit periodicity at that heartrate. Because thefetal heartbeat is generally non-synchronous with the maternalheartbeat, the fetal signal can be preserved and enhanced.

2.1 Baseline Wander Removal

Referring to FIGS. 2 and 3, in some embodiments, after acquisition ofthe raw electrode signals, baseline wander (BW) of the data is removedby a baseline wander system 200 (step 300) before further processing ofthe signals. In some examples, a two-stage moving window median (MWM)filter 202 with variable window lengths (for instance, set by default to200 ms and 600 ms, respectively) is used, followed by a lowpass filter204 with a variable cutoff frequency (for instance, set by default to 5Hz). The estimated BW obtained from filters 202 and 204 is subtractedfrom the input signal to achieve a BW-free signal. An additional lowpassfilter 206 with a variable cutoff frequency (for instance, set bydefault to 200 Hz) may also be used to suppress out-of-band noise.Alternatively, another effective BW removal strategy may be employed.

2.2 Cardiac Phase Signal

Cardiac waveforms have a pseudo-periodic structure that is repeated foreach cycle of the heartbeat but with normal RR-period deviations of upto 20% over the preceding RR interval (i.e., RR interval 358 in FIG.13). Referring to FIG. 4A, to represent this pseudo-periodicity for thecardiac source(s) of interest (i.e., the mother or the fetus), a cardiacphase signal Φ(t) is defined for a reference cardiac signal 400 receivedfrom an arbitrary reference channel (e.g., the maternal referencechannel). Φ(t) sweeps over [−π, π] for each cardiac cycle of referencesignal 400. Under this definition of Φ(t), samples of data havingsimilar phase values represent corresponding stages in thedepolarization/repolarization cycle of the heart and are morphologicallysimilar.

Referring to FIGS. 3 and 4A, to define Φ(t), R-peaks 404 of thereference cardiac waveform 400 are detected (step 302) using peakdetectors, matched filters, or another method capable of detectingR-peaks. Alternatively, R-peaks 404 may be detected using a time samplein which the cardiac beat has maximal ‘similarity’ with a givenreference beat. Based on R-peaks 404, Φ(t) is defined (step 304) as asaw-tooth signal 402 that linearly sweeps over [−π, π] in each R-Rinterval, with each R-peak 404 fixed at Φ(t)=0. Alternatively,non-linear mapping using, for instance, tanh functions may be used toconstruct Φ(t).

Based on Φ(t), a time-varying cardiac period τ_(t) 406 is defined asfollows:

τ_(t)=min{τ|Φ(t+τ)=Φ(t),τ>0}.

That is, the signal samples at time t and at time t+τ_(t) have the samephase value and are morphologically similar. τ_(t) is updated on abeat-to-beat basis. In general, the cardiac phase signal Φ(t) and thecardiac period τ_(t) may be defined for either the mother or the fetus,using the corresponding R-peaks.

The phase signal Φ(t) provides a means for phase-wrapping theRR-interval onto a [−π, π] interval. That is, an ECG, regardless of itsRR-interval deviations, can be converted to a polar representation inwhich the ECG components in different beats (e.g., the P, Q, R, S, andT-waves) are roughly phase-aligned with each other, particularly overthe QRS segment of the ECG. For instance, referring to FIG. 4B, a noisycardiac signal 450 is depicted in a polar representation of the ECGusing the phase Φ(t) of the signal.

2.3 Signal Extraction from a Scalar Input Signal

To demonstrate the general technique, one can consider a scalar case inwhich N=1. That is, there is a single abdominal electrode signal, andthe maternal heartrate is fixed with a period τ_(t). If the signal has avariance σ_(x) ² and a fixed autocorrelation at lag τ represented asR(τ)=E(x(t)x(t+τ))/σ_(x) ², then a denoising function can be applied tothe signal, for example based on the autocorrelation at lag τ. Anexample of a denoising function is a Weiner filter. Generally, for agreater autocorrelation value, more of the signal is removed asrepresenting a component that has periodicity that matches the maternalheartbeat.

In general, the maternal signal is not exactly periodic, and thereforeanalysis at a fixed lag may not be as effective as accounting for thevariation periodicity. That is, the period τ_(t) may vary as a functionof t. In this case, rather than an autocorrelation at a fixed lag, atime average at the time varying lag is used. For example, in the scalarcase, the non-stationary covariance {tilde over (C)}_(x), given as

{tilde over (C)} _(x) =E _(t)(x(t+τ _(t))x(t)),

is computed from the data, where E_(t)(•) is an average over time (e.g.,over a time window) that provides an approximation of theautocorrelation function. The time varying period may be determined, forexample, by tracking the R-R interval in the maternal reference signal.2.4 Signal Extraction from a Multidimensional Input Signal

In embodiments with multidimensional data (N>1) the scalar approachdescribed above is modified to include an additional feature ofidentifying a linear subspace that exhibits a relatively high degree ofsynchronization with the maternal heartbeat.

Specifically, rather than determining the sample variance of a scalarsignal, an N by N zero-lag stationary covariance matrix of the signalsis computed as:

C _(x) =E _(t) {x(t)x(t)^(T)},

where E_(t){•} represents an average over time (or expected value for astochastic formulation). Using the definition of the time-varyingcardiac period, the non-stationary covariance matrix at lag τ_(t) iscomputed as

{tilde over (C)} _(x) =E _(t) {x(t+τ _(t))x(t)^(T)},

which contains measures of the periodicity of the random process x(t)with respect to the heartbeat. That is, all of the temporal informationof the ECG is contained in {tilde over (C)}_(x). Furthermore, in someembodiments, {tilde over (C)}_(x) can be guaranteed to be positivedefinite (i.e., symmetric and with real eigenvalues) by applying

C _(x) ^(%)←(C _(x) ^(%) +C _(x) ^(%T))/2.

Considering the polar representations of each of the channels of x(t)(e.g., as shown in FIG. 4B), the matrix {tilde over (C)}_(x) representsthe covariance of the polar representation around its mean value. Thatis, {tilde over (C)}_(x) contains a measure of the ECG periodicityextracted from the ECG R-peak information.

In general, the cardiac phase signal Φ(t), the cardiac period τ_(t) andthe matrix {tilde over (C)}_(x) may be defined for either the mother orthe fetus, using the corresponding R-peaks.

In some embodiments, as described in greater detail below, the acquiredsignals are processed using an iterative deflation method for fetal ECGdenoising. The iterative method combines Pseudo-Periodic ComponentAnalysis (PiCA) and a Kalman denoising scheme for removing the maternalECG and enhancing the fetal ECG.

In some embodiments, as described in greater detail below, a generaldeflation method for subspace extraction is used to extract and enhancethe fetal ECG. The method uses an appropriate linear transform (such as,but not limited to, PiCA), with a linear or nonlinear denoising scheme(such as, Kalman filters, Wavelet decomposition, or etc.) for separatingany two intersecting subspaces of signal and noise.

2.4.1 Pseudo-Periodic Component Analysis (PiCA)

In some embodiments, a linear transformation based on generalizedeigenvalue decomposition (GEVD) of covariance and lagged-covariancematrices using time-varying lags is computed. The transformationgenerates linear combinations of the input data array, ranked in orderof resemblance with the periodicity of interest.

For instance, a linear subspace that exhibits relatively highsynchronization with the maternal heartbeat is identified based on thecomputed covariance and autocorrelation matrices. In some examples, aone-dimensional subspace is identified based on a generalizedeigenvector of the pair of matrices (C_(x), {tilde over (C)}′_(x))corresponding to the largest eigenvalue. Very generally, within theone-dimensional eigenspace, the scalar approach outlined above can beapplied to reduce the contribution of the maternal periodic componentaccording to the variance and autocorrelation values in thateigendirection.

More specifically, by GEVD of the matrix pair (C_(x), {tilde over(C)}_(x)), the matrices B and Λ are found such that

B^(T){tilde over (C)}_(x)B=Λ and B^(T)C_(x)B=I,

where Λ is the diagonal generalized eigenvalue matrix corresponding tothe generalized eigenmatrix B, with real eigenvalues sorted indescending order along its diagonal. B, a transformation thatsimultaneously diagonalizes C_(x) and {tilde over (C)}_(x), is referredto as the periodicity ranking transform (PRT). The first eigenvector b₁,which corresponds to the largest generalized eigenvalue, maximizes theRayleigh quotient:

${{J(b)} = \frac{b^{T}{\overset{\sim}{C}}_{x}b}{b^{T}C_{x}b}},$

which, following the definitions of C_(x) and {tilde over (C)}_(x), is ameasure of signal periodicity.

The transformed signals are then ranked in order of periodicity (i.e.,in order of relevance to the cardiac signal of interest). The rankedtransformed signals are represented as y(t), where

y(t)=B ^(T) x(t).

The components of y(t) (i.e., y(t)=[y_(l)(t), . . . , y_(n)(t)]^(T)) aresorted according to their periodicity such that y_(l) (t) is the mostperiodic component and y_(n) (t) is the least periodic with respect tothe R-peaks of the relevant ECG (step 306; also steps 500 and 550 inFIG. 5, discussed below).

For instance, for fetal ECG extraction, maternal and fetal ECG phasesfound from the maternal and fetal R-peaks may be defined as Φ_(m)(t) andΦ_(f)(t), respectively. The respective covariance matrices {tilde over(C)}_(x) ^(m) and {tilde over (C)}_(x) ^(f) can be found by averagingover the maternal and fetal periods. Then, the matrix {tilde over(C)}_(x) used in the GEVD process may be set to any of the followingmatrices:

{tilde over (C)}_(x)={tilde over (C)}_(x) ^(m)

{tilde over (C)}_(x)={tilde over (C)}_(x) ^(f)

{tilde over (C)} _(x) ={tilde over (C)} _(x) ^(m) −{tilde over (C)} _(x)^(f).

If we assume the data to be prewhitened, the diagonalization of theabove matrices is equivalent to finding, respectively, (1) the mostperiodic components with respect to the maternal ECG; (2) the mostperiodic components with respect to the fetal ECG; and (3) the mostperiodic components with respect to the maternal ECG while being theleast periodic components with respect to the fetal ECG. In this lattercase, the extracted components gradually change from the maternal ECG(the first components of y(t)) to the fetal ECG (the last components ofy(t)). The latter two cases may be difficult to implement in practice,as the extracted fetal R-peaks must be known in order to form the {tildeover (C)}_(x) ^(f) matrix.

In other embodiments, PiCA may be used to analyze twin fetal ECG (orMCG). In this case, after preprocessing of the data, the maternal ECG(or MCG) is removed using PiCA. An independent component analysis isused to find coarse estimates of the fetal ECGs or MCGs). The maternalECG (or MCG) is then reidentified using maternal/fetal PiCA as

{tilde over (C)} _(x) ={tilde over (C)} _(x) ^(m)−({tilde over (C)} _(x)^(f1) +{tilde over (C)} _(x) ^(f2)).

The fetal ECG (or MCG) is then determined in a post-processing step.PiCA is also extensible to use with multiple fetuses (e.g., triplets orhigher order multiples).

2.4.2 Denoising

Referring again to FIG. 3, the ranked signals y(t) are subjected toadenoising procedure, such as a wavelet denoising process. For instance,a multichannel extension of a Bayesian filtering framework is used todenoise the first M (1≦M≦N) components of y(t). For instance, a Kalmanfilter and Kalman smoother use a priori information from ECG dynamicsand noisy observations to estimate the true ECG and, optionally, toremove maternal ECG contaminants. Other, less optimal, denoising methodsmay also be used.

In the simplest case of M=1, the filtering is implemented as a singlechannel denoising.

In this framework, the components of y_(i)(t) (i=1 . . . M) areconsidered to be noisy observations of hidden states of a dynamicsystem. Each of the entries of the state vector is modeled with asummation of small waves, such as (but not limited to) Gaussianfunctions, which are pseudo-periodically repeated with the cardiac phasesignal Φ(t).

A state-space representation of the model is used for filtering; theparameters of the model are trained from the average and standarddeviations of the polar representation of the noisy signal of FIG. 4B(step 308). For online implementation of the filtering, the most recentcardiac beats are used in training the model and the parameters areupdated in real time.

Using this filtering framework, a denoised estimate {circumflex over(θ)}(t) of the cardiac signals is determined (step 310). If the maternalcardiac signals are of interest, the denoised cardiac signals are givenas z(t)={circumflex over (θ)}(t). If the residual non-maternal signalsare of interest (e.g., for removal of the maternal ECG/MCG), thedenoised signals are given as z(t)=y(t)−{circumflex over (θ)}(t) (step312).

The denoised signal z(t) is then transformed back to the sensor spaceusing the inverse PRT, producing a denoised version of x(t).

2.5 Iterative Subspace Removal

Subspace decomposition by deflation aims to decompose degeneratemixtures of signal and noise or artifacts without prior knowledge of thesubspace dimensions and without reducing the dimensions of the data.That is,

x(t)=x _(signal)(t)+x _(noise)(t)

and

y(t)=B ^(T) x(t)=B ^(T)(x _(signal)(t)+x _(noise)(t))=y _(signal)(t)+y_(noise)(t).

Subspace decomposition by deflation breaks the linearity by combiningsingle-channel denoising and multi-channel decomposition under theassumptions that (1) the desired signals in different channels aredependent such that processing gain is achieved through multichannelanalysis; and (2) some information is available about the desiredsignals such that the desired and undesired parts of the signals can beroughly separated using linear/nonlinear filters.

Referring to FIG. 5A, the process of projection (i.e., lineardecomposition of the input signals) using a periodicity rankingtransform (step 500), denoising using the Bayesian framework (step 502),and back-projection (i.e., linear recomposition of the denoised signals)using the inverse PRT (step 504) are repeated in one or severaliterations to gradually purify the observed signals. At each iteration,one eigendirection corresponding to the largest contribution of acontaminant signal (e.g., the maternal ECG) is identified and the effectof the contribution is reduced. This iteration may be performed a numberof times, for example, a fixed number of iterations, or based on adegree of any remaining maternal contribution at the output of eachiteration.

Referring to FIG. 5B, when applied to maternal ECG cancellation, anarray of signal recordings contaminated with maternal ECG are input intoa PiCA algorithm (550) along with the maternal ECG phase. The first Lcomponents output from the PiCA algorithm (i.e., the L components withperiodicity closest to the desired periodicity) undergo maternal ECGcancellation using a Kalman filter (552). The L filtered components andthe (N-L) remaining unfiltered components are recomposed by an inversePiCA algorithm (554). This step effectively removes any scaling andordering issues introduced by PiCA algorithms or other independentcomponent analysis (ICA) algorithms. The retransformed data isiteratively processed (556) until a recursion stopping criterion (558)is achieved, at which point data without maternal ECG (560) is output.

Referring to FIG. 6, the iterative subspace decomposition process isshown in more detail.

-   -   1. A set of input signals x^((k)) (t)=x(t) is received, where k        is an iteration index and is set initially to 0 (step 600).    -   2. The matrix C_(x) ^((k)), which is the covariance matrix of        x^((k))(t), is calculated (step 602).    -   3. The matrix {tilde over (C)}_(x) ^((k)), which contains the        desired statistics of x^((k))(t), is then determined (step 604).    -   4. B^((k)), which is the transpose of the decomposition matrix        found by generalized eigenvalue decomposition of the matrix pair        (C_(x) ^((k)), {tilde over (C)}^((k))) (i.e., B^((k))←GEVD(C_(x)        ^((k)), {tilde over (C)}_(x) ^((k)))^(T)), is obtained (step        606).    -   5. The transformed signals are ranked in order of periodicity,        giving y^((k))(t)←B^((k))x^((k))(t) (step 608).    -   6. The matrix A^((k))=&[a₁ ^((k)), . . . , a_(N) ^((k))]←B^((k))        ⁻¹ is defined (step 610), where a_(j) ^((k)) is the j-th column        vector of A^((k)).    -   7. A denoising function G(•) is applied to the first M channels        of y^((k))(t) to remove the undesired components and to keep the        desired components of each channel of y^((k))(t) (step 612). The        output of the denoising block in channel j is s_(j) ^((k))(t).        The overall output of the denoising function is s^((k))(t) =&[s₁        ^((k))(t), . . . , s_(M) ^((k))(t)]^(T) ←G(y^((k))(t)).    -   8. The updated signals x^((k+1))(t) of the iteration are        determined as x^((k+1))(t)←x^((k))(t)−Σ_(j=1) ^(M)a_(j)        ^((k))s_(j) ^((k))(t)(step 614).    -   9. ζ(x^((k+1))(t)), which is a measure of the desired subspace        removal used as a stopping criterion for the iterative subspace        decomposition algorithm, is calculated (step 616):        c←ζ(x^((k+1))(t)).    -   10. k is incremented: k←k+1 (step 618).    -   11. c is compared to a predefined threshold value th (step 620).        If c≦th, the iterative subspace removal process is stopped (step        622). Otherwise, a further iteration is performed (step 624).

The output of each iteration of the subspace removal algorithm can berepresented in compact form as

${x^{(k)}(t)} = {{x(t)} - {\sum\limits_{i = 0}^{k - 1}{\sum\limits_{j = 1}^{M}{a_{j}^{(i)}{{s_{j}^{(i)}(t)}.}}}}}$

The number of iterations performed depends on the dimensions of theremoved subspace. In general, for cardiac signals, between three to sixiterations are typically used. Thus, for practical implementations, thenumber of iterations may be fixed (e.g., at six iterations for cardiacapplications). Alternatively, in each iteration the projected signaly(t) may used to determine whether to continue or stop the subspaceremoval algorithm. Since the components of y(t) are ranked according totheir resemblance with the contaminating signal (e.g., the maternalECG), each iteration removes the signals that are least important to thedesired signal (e.g., the fetal ECG).

The following criterion may be used as an overall periodicity measure(PM) for x^((k))(t) in each iteration:

${{\zeta \left( {x^{(k)}(t)} \right)} = \frac{{trace}\left( {\overset{\sim}{C}}_{x}^{(k)} \right)}{{trace}\left( C_{x}^{(k)} \right)}},$

where C_(x) ^((k)) and {tilde over (C)}_(x) ^((k)) are the covariancematrices calculated for the signal x^((k))(t). Under this definition,0≦ζ(x^((k))(t))≦1, and the iterative subspace removal algorithm isstopped when the PM falls below a small predefined threshold 0≦th≦1 (forinstance, by default set at th=0.05).

Although only two matrices (C_(x) and {tilde over (C)}_(x)) are jointlydiagonalized, the results are still robust to deviations of heartbeatand noise. The reason is that the time lags τ_(t) required in thecalculation of {tilde over (C)}_(x) are extracted from the beat-to-beatinformation of the ECG. This robustness may be further improved bysplitting the information of the RR-interval that is carried by {tildeover (C)}_(x) into several matrices that contain local information aboutthe ECG cycle. Referring to the exemplary ECG cycle 350 shown in FIG.13, these matrices contain information such as the morphology of P-waves360 and T-waves 362, QRS complex 364, ST segment 366, QT interval 362,etc. These matrices can then be approximately diagonalized using jointapproximate diagonalization methods.

The iterative subspace removal algorithm is also effective when signalsare collected from a small number of electrodes. In other embodiments,when the number of observed channels is high and/or the desiredcomponents are strongly represented in the observed signals, oneiteration of periodicity ranking transformation, denoising of therelevant channels, and back-projection of the desired components may besufficient.

In some examples, a multidimensional subspace is identified at eachiteration, for example, corresponding to a number of the largestgeneralized eigenvalues. The number of generalized eigenvalues may befixed may be dependent on the data. The reduction in the maternalcomponent is effected according to the eigenvalue associated with thatdirection. In some examples, the full N-dimensional space is used ineach iteration.

2.6 Extraction of desired waveforms

Once the observed signals have been sufficiently purified as describedabove, the maternal cardiac interference is removed and the desiredfetal signals are extracted. In this case, rather than identifying thematernal heartbeats, the fetal heartbeats are identified in theremaining signal. In this enhancement phase, in some examples, asubspace, which exhibits periodicity at the fetal heartrate, isenhanced. As in the attenuation of the maternal signal, one or moreiterations may be performed, and at each iteration, one, a subset, orthe full N-dimensional space may be processed. Note that in someexamples, if the iterations of removal of the maternal signal arecontinued until the maternal signal no longer dominates the fetalsignal, the fetal periodicity may become evident without enhancementeven if it was not apparent in the original signal.

Referring to FIGS. 7A and 7B, for offline processing of batches ofsignals, the maternal cardiac interference is removed by a deflationprocedure (step 700) including detection of the maternal R-peaks (step702), calculation of the maternal cardiac phase signal (step 704), anditerative removal of the maternal cardiac signals (step 706). In oneembodiment, the fetal cardiac signal is then enhanced (step 708) by anylinear or nonlinear decomposition method. In another embodiment, thefetal cardiac signal is enhanced (step 710) by using a deflation methodtrained over the fetal R-peaks. That is, the fetal R-peaks are detected(step 712), the fetal cardiac phase signal is calculated (step 714), andthe fetal cardiac subspace is enhanced by an iterative estimationprocess (step 716).

Online implementation of the deflation procedure for the extraction offetal cardiac waveforms is similar to the offline batch method, exceptthat the PRT transform is extracted and updated in real time using themost recent cardiac beats (for either the mother or the fetus). Toimplement such real time processing, the covariance matrices C_(x)^((k)) and {tilde over (C)}_(x) ^((k)) are calculated online and updatedfrom sliding windows containing several maternal cardiac beats (e.g., 10beats), with 0%<n<100% of overlap (e.g., 25%<n<75%).

3 EXAMPLES Example 1

Referring to FIG. 8A, the well-known DaISy fetal ECG dataset (De Moor B.(ed.). DAISY: Database for the Identification of Systems, Dept. ofElectrical Engineering, ESAT/SISTA, K.U.Leuven, Belgium,URL:http://www.esat.kuleuven.ac.be/sista/daisy) is used to illustrate theabove method. The dataset includes five maternal abdominal channels andthree maternal thoracic channels recorded from the abdomen and chest ofa pregnant woman at a sampling rate of 250 Hz.

Referring to FIG. 8B, the maternal components are extracted from the ECGof FIG. 8A. Specifically, by performing R-wave detection on one of thematernal thoracic channels, the maternal ECG phase Φ_(m)(t) isdetermined. The time-varying maternal ECG period τ_(t) ^(m) iscalculated, from which the matrix {tilde over (C)}_(x) and thegeneralized eigenmatrix B of the (C_(x) ^(%), C_(x)) pair are found andsorted in descending order of the eigenvalues. The resulting periodcomponents extracted from the dataset by maternal ECG beatsynchronization are shown. The first component (curve 800), whichcorresponds to the largest eigenvalue, has the most resemblance with thematernal ECG. As the eigenvalues decrease (i.e., curves 802-814), thesignals become less similar to the maternal ECG. Curves 810 and 812 arethe fetal components. Because the fetal components do not resemble thematernal ECG when averaged synchronously with respect to the maternalR-peaks, the fetal components are among the last components extracted.

Referring to FIG. 8C, the fetal components are now extracted from theECG of FIG. 8A. ECG periodicity is considered in the matrix {tilde over(C)}_(x), which uses the fetal R-peaks for extraction of thetime-varying fetal period τ_(t) ^(f). In this example, the fetal ECGcomponent is first extracted from the dataset of FIG. 8A by a benchmarkindependent component analysis method known as JADE. The extracted fetalECG component is used for fetal R-peak detection and phase calculation.Having determined the fetal ECG phase Φ_(f)(t), the GEVD processdescribed above is used to extract the periodic components of the fetalECG. The resulting periodic components are depicted in FIG. 8C, with theextracted components ranked according to their resemblance with thefetal ECG (i.e., the fetal ECG contribution reduces from top to bottom).

Referring to FIG. 8D, periodic components are extracted from the datasetusing maternal and fetal ECG beat synchronization. The covariance matrix{tilde over (C)}_(x) is calculated from the difference of {tilde over(C)}_(x) ^(m) and {tilde over (C)}_(x) ^(f). The periodic components aredetermined as described above. The first component (curve 850) includesprimarily maternal ECG; the last component (curve 860) includesprimarily fetal ECG. The intermediate components are mixtures ofmaternal and fetal components and noise.

Example 2

Referring to FIGS. 9A-9F, the fetal components of the DaISY datasetshown in FIG. 9A are extracted and the maternal components removed by aniterative subspace removal process. Each of FIGS. 9B-9F corresponds tothe result of one iteration. The periodicity measure of each iterationis listed in Table 1. The maternal ECG contaminants have beeneffectively reduced from the first to the last iteration. Referring toFIG. 10, a secondary deflation step was applied to the results of thelast iteration of maternal ECG removal to enhance the fetal ECG. Thefetal ECG signals before enhancement (gray line) and after enhancement(black line) are shown superimposed.

Example 3

Referring to FIG. 11, in another example, data were acquired from 27term laboring women who had a fetal scalp electrode (FSE) placed for aclinical indication. A total of 27 channels of abdominal data and onechannel of data from a chest lead were recorded simultaneously with theFSE. The median ST level was estimated from 79 10-second segmentsacquired from the FSE and from preprocessed abdominal data which passedan unbiased quality test. A statistical comparison was performed toassess the accuracy of the ST deviation derived from abdominal sensorsas compared to the ST deviation obtained from the “gold standard,” FSE.FIG. 11 illustrates a typical fetal heart beat recorded from the scalpelectrode and an extracted abdominal fetal ECG overlaid. The beatmorphology is well reconstructed by the extracted signal and theextracted fetal ECG shows good fidelity with the fetal scalp ECG. Inparticular, the morphologies of all the clinical features are wellpreserved in the extracted abdominal fetal ECG, particularly the STsegment deviation. The QRS is also accurately extracted from theabdominal channels, including the R-peak height and location (relevantfor respiration and heart rate (HRV) analysis, respectively). The P-waveand repolarization periods are also accurately extracted compared to thescalp electrode.

When comparing the difference in the ST height above the isoelectricline between the fetal scalp electrode signal and the signal extractedfrom the abdominal electrodes, the root mean square error (difference)over all processed segments was determined to be on average 0.52% andthe mean absolute difference was 0.29%. These units are expressed inpercent relative to the R-peak height because the electromagnetic fieldstrength is different on the abdomen and the fetal scalp. Moreover, theST elevation for fetuses is defined in terms of its relative heightcompared to the R-peak. To better understand these statistics, we cancompare to an adult ECG. A 0.52% deviation of an adult ST segment on a1.5 mV amplitude beat corresponds to 0.0078 mV, which is well below theacceptable error level and would not lead to a false identification ofST elevation (or depression). FIG. 12 illustrates the distribution ofthe difference in ST deviation between the fetal scalp ECG and extractedabdominal fetal ECG. The gray line 1200 indicates a Gaussian fit to theempirically calculated distribution (vertical bars). Even at the extremevalues of 2% deviation, the equivalent adult ST deviation would only be0.03 mV, which is not clinically significant when compared to theamplitude of an adult ECG.

Therefore, the fetal ST deviation calculated from ECG acquired from thematernal abdomen on healthy subjects is clinically indistinguishablefrom ST deviations derived from the fetal scalp electrode when using theextraction method detailed in this application.

TABLE 1 Periodicity measure variation with respect to maternal ECG foreach of five iterations Iteration 1 2 3 4 5 ζ(x^((k)) (t)) 0.98 0.910.51 0.34 0.28

In general, the methods described above are applicable in clinical,theoretical, and experimental situations. For instance, clinical studyof pathological cases is enabled. Theoretically, performance bounds forECG processing can be calculated and theoretical aspects of thedeflation method (e.g., convergence and stability) can be studied.Experimentally, fetal ECG tracking for continuous monitoring can beinvestigated and fetal monitoring systems can be developed.

4 IMPLEMENTATIONS

The approaches described above may be implemented in hardware, software,or using a combination of hardware and software. Hardware may includeanalog signal processing circuitry, digital signal processing circuitryor both. Software may include instructions stored on or transmitted overa machine readable medium. The instructions can include instructions forcausing a processor (e.g., a general purpose computer processor, adigital signal processor, special purpose processor, virtual computer, acomplex programmable logic device (CPLD), a field programmable gatearray (FPGA), etc.) to implement the processing steps of approachesdescribed above. The instructions can include instructions for multipleseparate processors, which may be integrated into a single unit ordistributed into multiple separate units.

In some implementations, a system includes signal acquisition hardware,which can include components for receiving biosignals such as ECGsignals, and can include the sensors that acquire such signals, such aselectrodes for placement on the maternal abdomen. In someimplementations, a system includes signal presentation hardware, whichcan include components for providing processed biosignals forpresentation or for further processing prior to analysis orpresentation. In one implementation, the processed biosignals (e.g.,fetal ECG) are presented on a computer generated display for monitoringby medical staff. In some instances, the processed biosignals are usedto determine an orientation of the fetus and data or imagesrepresentative of the fetal orientation are presented on a computergenerated display. As another example, the processed biosignals may beprovided to a system that analyzes extracted fetal ECG signals asdescribed in copending U.S. Patent Pub. US2009/0259133A1, titled “FETALECG MONITORING.” Data representative of the analysis, such as adiagnostic score related to the entropy (i.e., variability) of theprocessed biosignals, are presented on a computer generated display.

Although primarily described with reference to maternal ECGcancellation, the methods disclosed herein are broadly applicable to awide range of data processing situations in which a contaminant signalis to be removed. For instance, ECG artifacts may be removed from otherbiosignals such as an electroencephalogram (ECG) or an electromyogram(EMG). More generally, enhancement and/or removal of pseudo-periodicsignals may be applied in various areas including communication andsensor systems. As one example, communication signals that includeperiodic power line interference may be enhanced using the techniquesdescribed above, for example, for mitigation of pseudo-periodicinterference in communication signals over powerlines.

It is to be understood that the foregoing description is intended toillustrate and not to limit the scope of the invention, which includesthe scope of the appended claims. Other embodiments are within the scopeof the following claims, and other aspects are not necessarilyrepresented by the following claims.

1. A method for processing signals comprising: accepting, from a sensorsystem, a set of one or more signals, the signals including componentsof a desired signal and components of a interfering signal having aperiodicity; determining, by a processing engine, the periodicity of theinterfering signal; and performing, by the processing engine, one ormore iterations of mitigating an effect of a component of the signalsthat exhibits periodicity at the determined periodicity of theinterfering signal to form a processed signal.
 2. The method of claim 1further comprising: providing, to a display system, the processedsignal.
 3. The method of claim 1 wherein the interfering signal includesat least one of a substantially periodic and a pseudo periodic signal.4. The method of claim 1 wherein determining the periodicity of theinterfering signal includes determining a time varying period of saidsignal.
 5. The method of claim 1 further comprising: determining aperiodicity of the desired signal; and performing one or more iterationsof enhancing an effect of a component of the signals that exhibitsperiodicity at the determined periodicity of the desired signal.
 6. Themethod of claim 1, wherein the desired signal is a cardiac signal. 7.The method of claim 6 wherein the interfering signal comprises a secondcardiac signal.
 8. The method of claim 7, wherein the periodicity of theinterfering signal comprises a quantity characterizing a heartbeatperiod of the second cardiac signal.
 9. The method of claim 8 whereinthe quantity characterizing the heartbeat period of the second cardiacsignal comprises an estimated R-R interval.
 10. The method of claim 6wherein the desired cardiac signal comprises a fetal cardiac signal andwherein the interfering signal comprises a maternal cardiac signal. 11.The method of claim 6, wherein the desired cardiac signal comprises acardiac signal of a first fetus and the interfering signal comprises acardiac signal of a second fetus.
 12. The method of claim 11, whereinthe interfering signal further comprises a maternal cardiac signal. 13.The method of claim 1 wherein mitigating the effect of a component ofthe signals that exhibit periodicity at the determined periodicity ofthe interfering signal includes: identifying a relative contribution ofthe component of the signals that exhibit periodicity at the determinedperiodicity; and processing the signals according to the identifiedrelative contribution.
 14. The method of claim 13 wherein identifyingthe relative contribution of the component of the signals that exhibitperiodicity at the determined periodicity includes computing ageneralized eigenvalue based on correlation of the signals at thedetermined periodicity of the interfering signal.
 15. The method ofclaim 13 wherein processing the signals according to the identifiedrelative contribution includes applying a Bayesian filtering approach.16. A cardiac monitoring system comprising: an input subsystem foraccepting signals, the signals including components of a desired signaland components of an interfering signal having a periodicity; a dataprocessing system for processing the accepted signals, the dataprocessing system being configured to perform the steps of: determiningthe periodicity of the interfering signal, and performing one or moreiterations of mitigating an effect of a component of the signals thatexhibits periodicity at the determined periodicity of the interferingsignal; and a data output subsystem for providing information determinedfrom the processed signals.
 17. The cardiac monitoring system of claim16, wherein the desired signal is a cardiac signal.
 18. Softwareembodied on a computer-readable medium comprising instructions forcausing a data processing system to accept a set of one or more signals,the signals including components of a desired signal and components of ainterfering signal having a periodicity; determine the periodicity ofthe interfering signal; and perform one or more iterations of mitigatingan effect of a component of the signals that exhibits periodicity at thedetermined periodicity of the interfering signal.
 19. A method forprocessing a input signal that includes a pseudo-periodic signal presentwith another signal, the method comprising: accepting the input signal,the input signal being representable as a vector signal in amultidimensional vector space; forming a phase signal representing atime-varying periodicity of the pseudo-periodic signal; identifying asubspace of the vector space that exhibits correlation at thetime-varying periodicity; forming a component of the input signalaccording to the identified subspace; and processing the input signal toeither enhance or to mitigate and effect of the formed component toproduce a processed signal.
 20. The method of claim 19 wherein acceptingthe input signal comprises accepting a biosignal, the pseudo-periodicsignal having a periodicity related a periodic biological process. 21.The method of claim 20 wherein the biosignal comprises signal acquiredby sensing a signal from a human body, and wherein the periodicbiological process comprises a cardiac process.
 22. The method of claim19 further comprising presenting the processed signal on a display. 23.The method of claim 19 wherein forming the phase representation includesidentifying repetitions of a signal characteristic in the input signal;identifying the subspace includes determining a generalized eigenspacefrom a correlation of the input signal at a time varying lag determinedfrom the phase representation and from a correlation of the input signalat zero lag; forming the component of the input signal according to theidentified subspace includes projecting the input signal onto theeigenspace; and processing the input signal to either enhance or tomitigate and effect of the formed component to produce a processedsignal includes at least one of adding or subtracting the projection ofthe signal onto the eigenspace.